Trajectory optimization for the Horyu-VI international lunar mission
Federico De Grossi1(B), Paolo Marzioli1, Mengu Cho2, Fabio Santoni1, and Christian Circi1
1. Sapienza University of Rome, 00138 Rome, Italy
2. Kyushu Institute of Technology, Kitakyushu 804-8550, Japan
ABSTRACT KEYWORDS
The Horyu-VI nano-satellite is an international lunar mission with the purpose of studying the lunar horizon glow (LHG)—a still unclear phenomenon caused by electrostatically charged lunar dust particles. This study analyzes the mission trajectory with the hypothesis that it is launched as a secondary payload of the NASA ARTEMIS-II mission.
In particular, the effect of the solar gravity gradient is studied; in fact, depending on the starting relative position of the Moon, the Earth, and the Sun, the solar gradient acts differently on the trajectory—changing it significantly. Therefore, the transfer and lunar capture problem is solved in several cases with the initial Sun–Earth–Moon angle as the key parameter. Furthermore, the inclination with respect to the Moon at capture is constrained to be equatorial. Finally, the problem of stabilization and circularization of the lunar orbit is addressed in a specific case, providing an estimate of the total propellant cost to reach the final orbit around the Moon.
trajectory optimization Artemis mission lunar mission
lunar horizon glow (LHG) weak stability boundary
Research Article Received: 30 April 2021 Accepted: 2 July 2021
©The Author(s) 2021
1 Introduction
The Horyu-VI mission is a nano-satellite lunar mission developed by the Kyushu Institute of Technology, in col- laboration with the Nanyang Technological University of Singapore, Sapienza University of Rome, and the Cali- fornia Polytechnic State University [1]. The mission was proposed as a secondary payload of the ARTEMIS-II NASA mission.
The purpose of Horyu-VI is to study the lunar horizon glow (LHG) phenomenon. It appears as a slim bright- ness above the lunar surface, and was observed by the astronauts of the Apollo missions and pictures of it were taken by Surveyor 1, 5, 6, and 7. It is believed that the LHG is caused by lunar dust particles scattering light at sunrise and sunset; and since the brightness was stronger than the level that could be produced by micrometeorite ejecta, a mobilized dust population by electrostatic forces was predicted to be responsible for the forward scattering of sunlight [1–12].
To date, there has been no dedicated mission to ob- serve the lunar horizon glow, especially near and be-
low a 1-km altitude. A small satellite platform can be used to continuously monitor the forward-scattering of sunlight above the lunar terminator region to de- tect the column density of the lunar dust cloud. The most recent attempt to detect the lunar horizon glow came from the lunar orbiter laser altimeter and the laser ranging telescope onboard the Lunar Reconnais- sance Orbiter [2]. These measurements aimed to test the hypothesis of a major meteor stream producing sufficient ejecta for the LHG. The outcome is most likely that the LHG is a rare occurrence at altitudes below 20 km during meteor streams. These measure- ments could not put any constraint for the LHG near a 1-km altitude and below—where most of the exploration activities are present. The study of the LHG could reveal critical information about the dust particles near the surface and at low altitudes—information which could be important for future human and robotic missions since the regolith dust is a significant risk factor in the lu- nar environment [13–18]. A low lunar orbit is the most promising candidate for studying the LHG, from where Bfederico.degrossi@uniroma1.it
Nomenclature
a semi-major axis (km)
e eccentricity
i inclination (◦)
Ω RAAN (◦)
ω argument of pericenter (◦)
θ true anomaly (◦)
r spacecraft position vector (km)
v spacecraft velocity vector (km/s)
rm position vector from the Moon to the spacecraft (km)
Rm Moon position vector (km)
rs position vector from the Sun to the spacecraft (km)
Rs Sun position vector (km)
F thrust of the spacecraft (N)
m mass of the spacecraft (kg)
Isp specific impulse (s)
µE Earth gravitational constant (km3/s2)
µM Moon gravitational constant (km3/s2)
µS Sun gravitational constant (km3/s2)
θsm0 Sun–Earth–Moon angle at the initial time (◦)
J cost function
tf time of flight (day)
mp mass of propellant (kg)
α, β angles of the thrust—variables of optimization (◦)
Narc number of arcs of a trajectory
hp lunar pericenter height (km)
hflyby height of lunar flyby (km)
aSRP solar radiation pressure acceleration vector (m/s2)
A spacecraft area (m2)
PE solar radiation pressure at 1 AU (N/m2)
it is also possible, in principle, to extend the scientific objectives by releasing femto-satellites—such as smart dust from the satellite [19].
The present work focuses on the trajectory analysis of the Horyu mission: the transfer trajectory, in particular the effect that the solar gravity gradient has on it, and the stabilization of the orbit around the Moon.
The spacecraft is considered equipped with low-thrust Hall effect thrusters, whose acceleration is smaller than the Sun gravity gradient for most of the time during the transfer trajectory to the Moon; therefore, the effect of the Sun is of great importance and, according to the initial position of the Earth and the Moon with respect to the Sun, the trajectory changes significantly. The trajectory is computed in several cases, covering all the variation range of the Sun–Earth–Moon angle, which is the main parameter that changes the effect of the solar gradient on the trajectory.
The paper is composed of the following sections:
Section 2 presents the spacecraft architecture and main design features. In Section 3, the strategy employed for the design and optimization of the trajectory is described.
In Section 4, the obtained results are presented and analyzed, and a constraint on the final inclination at the arrival at the Moon is considered, as well as its influence on the trajectory. Section 5 presents further analysis on the sensitivity of the trajectory to small changes in the initial position of the Moon, affecting the flyby at the beginning of the trajectory, as well as an estimate of the effect of the solar radiation pressure.
In Section 6, the problem of stabilization and circular- ization of the orbit is considered for one of the obtained trajectories. The conclusions are finally presented in Section 7.
2 Horyu-VI spacecraft architecture and design features
The Horyu-VI nano-satellite is a 12U CubeSat (approx- imately 200 mm ×200 mm × 300 mm) conceived for LHG observations in a lunar trajectory [1]. The space- craft architecture is illustrated in Fig. 1. The mission was conceived for NASA’s ARTEMIS-II call for secondary payload proposals issued in 2019. The CubeSat payload
Fig. 1 CAD image of Horyu-VI.
is composed of imaging sensors at different wavelengths (monochrome, RGB, near-infrared (NIR) and ultraviolet (UV) to detect and investigate LHG in lunar orbit. The spacecraft navigation will be made possible by ground- based range and range rate determination. The on-board transponder uses a chip-sized atomic clock (CSAC) for time reference. The cis-lunar space characterization of the behavior of such a time-keeping device, whose oper- ations were already in-orbit demonstrated through the SPATIUM-I mission in low Earth orbit (LEO), can be considered a secondary objective for spacecraft missions.
The spacecraft propulsion system relies on four xenon gas Hall effect micro-thrusters capable of providing a thrust of 150µN each. The thrusters are located on the same side of the spacecraft, providing a total thrust of 600µN. The required electrical power for the maneuvers, approximately 60 W for the entire system, will be gener- ated by the spacecraft deployable solar panels and stored by the on-board batteries. The total amount of stored gas propellant is approximately 3 kg; and the dedicated CubeSat volume to the propulsion and propellant storage subsystems was approximately 8 units. The spacecraft propulsion system does not involve any thrust vectoring systems, so the thrust vector direction is controlled by the attitude determination and control subsystem. This will rely on reaction wheels for fine attitude control and on cold-gas thrusters, using xenon and sharing the propel- lant tanks with the propulsion system for de-saturation.
The main attitude determination sensors are optical star trackers, and inertial measurement units. The entire system will assure a 0.1 degree pointing accuracy.
3 Trajectory design
As stated above, the mission begins as a secondary pay- load of the SLS rocket in an Artemis mission. The SLS upper stage can release the secondary payloads on three bus stops along its disposal trajectory at altitudes of
36,500, 70,000, and 93,500 km [20]. In this mission anal- ysis, it is assumed that the spacecraft is released at an altitude of 70,000 km on the upper stage disposal trajec- tory. If no thrust is applied, the spacecraft encounters the Moon on a flyby—which increases its energy; there- fore, without control, the spacecraft will abandon the Earth–Moon system.
The initial conditions for our simulation are in reference to December 15, 2017–14:56:42.2 TDB date; and the trajectory is propagated considering the Earth–Moon–
Sun restricted four-body problem dynamics. The initial orbital elements, referred to as the Earth, are given in Eq. (1); and the equations of motion of the spacecraft in a Cartesian Earth-centered inertial reference frame are given in Eq. (2).
(a= 206076.92 km, e= 0.9667, i= 28.61◦
Ω = 65.96◦, ω= 47.92◦, θ0= 148.41◦ (1)
˙ r=v
˙
v=gE+gpM+gpS+ac =−µE
r3r
−µM
rm
rm3 +Rm
R3m
−µS
rs
rs3+Rs
R3s
+F m
(2) wherer,v are the position and velocity vectors respec- tively,gE is the gravitational acceleration of the Earth, gpM is the gravity effect from the Moon,gpS is the grav- ity effect from the Sun; ac is the thrust acceleration, µE, µM, and µS are the Earth, the Moon, and the Sun gravitational parameters, respectively;rm is the vector from the Moon to the spacecraft;Rmis the vector from the Earth to the Moon;rsis the vector from the Sun to the spacecraft; Rs is the vector from the Earth to the Sun;F is the thrust vector; and finally,mis the mass.
The spacecraft is supposed to have an initial mass m0= 20 kg, a total thrust ofF= 600µN, and a specific impulse ofIsp = 1000 s.
The final orbit around the Moon is a circular, equatorial low lunar orbit with a height of 100 km, and there are no further constraints on the right ascension or the argument of the pericenter. For the purpose of this study, the orbit to be achieved can be prograde or retrograde, with a margin of 30◦from the conditions of the equatorial orbit.
In the following section, only the transfer trajectory is addressed, with the objective of obtaining an orbit with eccentricity less than one, and pericenter height equal to 200 km. Initially, the inclination of the arrival orbit
around the Moon was left as free, and then the difference was shown when it was constrained instead.
Different launching dates for the mission imply differ- ent Sun–Moon relative initial positions; therefore, the solar gradient affects the trajectory in different ways—
changing the cost of the trajectory in terms of time and propellant needed. To analyze these effects, starting from the configuration of the date to which the initial condi- tions are referred to, the position of the Sun is moved by intervals of 10◦ of the true anomaly along its ap- parent orbit around the Earth, until a complete turn is completed. This generates new configurations of the Sun–Earth–Moon system, which can be described by the Sun–Earth–Moon angle at the initial timeθsm0; and the trajectory is therefore computed in all 36 configurations ofθsm0.
θsm0=±arccos(Rbm0·Rbs0) (3) The Sun–Earth–Moon angle is computed as in Eq. (3), and the plus sign is taken for the anti-clockwise rotation of Rbs0fromRbm0, and the minus otherwise. Figure 2 shows the angle, showing the orbital planes of the Moon and Sun in the Earth-centered frame.
The problem is approached as an optimization problem.
The trajectory is divided in N arcs of duration ∆ti, during which the thrust direction is fixed in the inertial frame and defined by two angles αi and βi, and the number of optimization variables is therefore 3N.
Fi=Fnom(cosαicosβi sinαicosβi sinβi)T (4) uopt= (∆t1∆t2· · ·∆tN,α1β1α2β2· · ·αNβN)T (5)
J=tf+C (6)
Equation (4) is the thrust vector in the inertial geocen- tric frame during theith arc; Eq. (5) shows the vector of optimization variables; Eq. (6) is the cost function, where tf =PN
i=1∆ti is the duration of the trajectory, and C is the constraint on the final state as penalty functions.
No coasting arcs are considered; therefore, if the tf is minimum, the propellant mass is also minimum.
The design method of the trajectory is as follows: a first starting guess is obtained by manually choosing the variables, and then an optimization is run with C in the cost function in the form of Eq. (7)—whererj and Rmj are the components of the position vector of the spacecraft and of the Moon in the geocentric frame re- spectively. vj andVmj are the components of the velocity vector of the spacecraft and of the Moon in the geocentric frame respectively, as in common rendezvous trajectory approaches [21]. The first optimization aims to find a trajectory in which the spacecraft encounters the Moon.
However, it is not sufficient to obtain the orbit with the required pericenter height as it often yields final condi- tions with non-optimal semi-axis and eccentricity—which can be improved. Therefore, another optimization is run withC as shown in Eq. (8), wherehp is the pericenter height over the Moon, ht = 200 km, and em and am are the final eccentricity and semi-axis with respect to the Moon respectively. The optimization is repeated, progressively reducingatandet, the target semi-axis and eccentricity around the Moon respectively, untilamand emno longer decrease, withem<1 andhp=ht.
C=cRm 3
X
j=1
|rj−Rmj|+cVm 3
X
j=1
|vj−Vmj| (7)
[
Z Y
3 3
Fig. 2 Visualization of the Sun–Earth–Moon angleθsm0.
C=chp|hp−ht|+ce|em−et|+ca|am−at| (8) The employed optimizers are the interior-point al- gorithm of the MATLAB function fmincon; the con- straints considered are only the boundaries: ∆ti,min = 0,∆ti,max = 400 days, αi,min = −180◦,αi,max = 180◦, βi,min=−90◦, βi,max= 90◦; the remaining options were set as default, except for the maximum function evalu- ations, which were set at 4000. The Nelder–Mead algo- rithm of the functionfminsearch was used as well, with default options except the maximum number of func- tion evaluations, which were set at 3000. The MATLAB version used was R2019a.
4 Discussion on the results
The exploitation of the solar gradient to obtain low-cost Earth–Moon transfers is employed in the weak stabil- ity boundary (WSB) or Belbruno lunar trajectories. In these trajectories, the gravity gradients of the Sun and the Earth are used to modify the trajectory, raising the perigee in the intermediate phase and decelerating in the final phase to obtain a ballistic capture around the Moon [22–24]. WSB trajectories were used in the Earth–
Moon transfers by the JAXA Hiten mission (1990) for the first time, as suggested by Edward Belbruno and
later by the NASA GRAIL mission (2011). A similar strategy can be exploited in our case where we are trying to obtain the best effect from the Sun and the Earth to achieve a capture similar to the one typical of WSB. In contrast to the classical ballistic WSB trajectory in our problem, there is also a flyby, and low-thrust propulsion is used.
After the lunar flyby, the spacecraft needs to be de- celerated—but the thrust alone is not sufficient to do this, especially if the solar gradient accelerates instead.
The analysis showed that the most efficient trajectories have an initial phase before the apogee in which the solar gradient decelerates the spacecraft, and a phase after the apogee where it accelerates, easing the increase of the perigee. This class of trajectories are the best in terms of time of flight and propellant among the ones found, and appear to be possible for values approximately between θsm0∈[−30◦,0◦], and betweenθsm0∈[160◦,180◦]—with a duration between 81 and 141 days. Figure 3 shows one such trajectory in the geocentric reference frame, the thrust (blue arrows) and the gradient (red arrows) are shown; the red dashed line represents the scaled path of the Sun during the trajectory. The trajectory showed is the best among the ones found; it has atf= 81.5 days, a mass of propellant required of 0.43 kg, and it achieves a lunar orbit with an eccentricity of 0.93, and the value of
x
y
Trajectory
Fig. 3 WSB-like trajectory in the Earth-centred reference frame,θsm0=−5.78◦.
the Sun–Earth–Moon angle is equal toθsm0=−5.78◦. Figure 4 shows the same trajectory in the Sun–Earth co-rotating frame in the left plot. In this reference frame, the direction of the Sun is constant; thus, the gradient acceleration field does not change—it is represented by the red arrows in the plot. The right plot shows the arrival at the Moon. It can be seen that the Earth’s gravity gradient decelerates the spacecraft in this final phase, helping during the capture. The expressions of the gradient accelerations of the Sun and Earth plotted in Figs. 3 and 4 are given in Eq. (9):
gS=−µS
rs r3s +Rs
R3s
, gE=−µE
re re3 +Re
R3e
(9) As before, rs and re are the vectors from the Sun to the spacecraft and from the Earth to the spacecraft, respectively; andRs, Re are the vectors from the Earth to the Sun and from the Moon to the Earth, respectively.
In Fig. 3, the gradient is given by the expression on the left in Eq. (9) in the Earth-centered frame, where Rs varies according to the motion of the Sun with respect to the Earth. In Fig. 4 on the left, gS is computed in a reference frame with thex-axis parallel to the Earth–
Sun direction; therefore, Rs is constant in this frame.
In Fig. 4 on the left, the Earth gradient is given by the expression on the right in Eq. (9) in a Moon-centered non-rotating frame, whereRe rotates according to the
motion of the Earth with respect to the Moon.
Figure 5 shows a comparison plot of the norm of the thrust acceleration and the norm of the Sun gradient acceleration during the trajectory of Figs. 3 and 4. It can be seen that the solar gradient is a dominating ac- celeration in the central part of the trajectory, which is the longest part of it.
For values ofθsm0farther from the most favorable ones, the trajectories have a farther apogee because the Sun gradient decelerates less after the flyby; consequently, the trajectory requires more time and more thrust effort to avoid escaping the Earth at the apogee. Trajectories like Fig. 6 can be found, presenting an unstable behavior at the apogee, and the angles of these solutions are the limit values for which WSB-like trajectories are found, after which the resemblance with WSB trajectories disappears.
In the cases where the Sun gradient accelerates the spacecraft after the flyby, the thrust is not sufficient to avoid the geocentric energy becoming greater than zero;
in these cases, the duration of the trajectory increases significantly, and the thrust can be used to allow the spacecraft to re-encounter the Earth in a favorable con- dition for the Moon rendezvous. An example is shown in Fig. 7, where the entire trajectory is shown in the left plot, and it is zoomed on the Earth in the right plot to better show the departure and arrival.
Z Z
Y Y
Fig. 4 WSB-like trajectory in the Sun–Earth rotating frame and at the arrival at the Moon.
Fig. 5 Norm of the solar gradient acceleration and norm of the thrust acceleration during the trajectory of Fig. 3.
Y
Z
Fig. 6 Trajectory withθsm0= 27.5◦, tf = 165.9 days. Earth-centered inertial reference frame.
The values of the optimization variables for the three trajectories presented above are listed in Tables 1, 2, and 3.
Overall, for all the values ofθsm0 considered, it was
possible to find a trajectory that achieved a close orbit around the Moon; the fastest trajectory is the one in Fig. 1 forθsm0=−5.78◦ withtf = 81.5 days andmp= 0.43 kg. The longest trajectory hastf= 1038.8 days, and
Y Y
ZZ
Fig. 7 Trajectory withθsm0=−63◦, tf = 468.1 days. Earth-centered inertial reference frame.
Table 1 Design variable values for the trajectory with θsm0=−5.78◦
Number of arcs ∆t(day) α(deg) β(deg)
1 2.849 112.106 11.059
2 18.899 129.573 0.563
3 19.989 124.637 0.009
4 22.069 131.268 48.833
5 10.607 172.478 69.616
6 7.126 159.338 0.099
Table 2 Design variable values for the trajectory with θsm0= 27.5◦
Number of arcs ∆t(day) α(deg) β(deg)
1 16.886 129.248 21.155
2 11.009 113.292 27.056
3 21.104 −104.280 18.406
4 42.540 51.244 48.687
5 49.601 73.611 0.435
6 6.459 −53.544 7.504
7 10.650 −77.522 −4.959
8 7.699 −176.076 5.307
mp = 5.47 kg, and it is found for θsm0 = 67.21◦. The trajectories have a number of arcsN between 6 and 25.
The best trajectories generally needed a lesser number of arcs, while the longest trajectories required more arcs.
The lunar orbits at arrival have eccentricities between e= 0.927 ande= 0.963. Figure 8 shows the plots of the time of flight, propellant to initial mass ratio mp/m0, and the height of the flyby with the Moon for all the values of the angleθsm0considered. Minima can be seen aroundθsm0= 0◦,180◦; and it can be noted that they roughly correspond to the maxima of thehflyby, which
Table 3 Design variable values for the trajectory with θsm0=−63◦
Number of arcs ∆t(day) α(deg) β(deg)
1 4.112 −169.676 57.919
2 14.176 −71.686 −22.700
3 27.985 −69.839 −27.865
4 7.112 −48.758 −8.150
5 9.365 −43.015 −24.340
6 23.736 −14.619 −10.417
7 23.755 7.479 −5.122
8 64.520 −11.906 −54.167
9 65.219 170.868 30.101
10 14.117 137.646 16.012
11 54.853 −163.998 −35.248
12 47.469 −160.880 −17.451
13 42.336 −103.778 −7.758
14 38.809 −55.258 −12.083
15 25.368 −54.266 −21.555
16 5.182 43.908 14.801
resulted in variations between 829 and 3200 km.
Figure 9 shows the behavior of the ∆V with respect to θsm0. The ∆Vtrajis the effective ∆V exerted during the trajectory, and its behavior is similar to that ofmp/m0
andtf, with the same maxima and minima. The ∆Vcirc
is the impulsive ∆V that should be exerted to circularize the lunar orbit at an altitude of 200 km, and it does not represent the true circularization effort—which will need to be carried out with low-thrust propulsion. The only thing worth noting from it is that the orbits achieved are quite similar as the difference in ∆Vcirc between one another is just some tens of m/s. The ∆Vtot in the final plot is the sum of the previous two, and includes the departure ∆Vdep given by the launcher to put the
Fig. 8 Time of flight (tf), propellant mass ratio (mp/m0), and flyby height (hflyby).
Fig. 9 Trajectory ∆V, Circularization impulsive ∆V, and total ∆V. payload in lunar transfer from a circular orbit around
the Earth—which is estimated to be 3068 m/s.
In both Fig. 8 and the first and last plots of Fig. 9, it can be seen that the solutions exhibit some periodicity
after 180◦ of variation of the angle parameter. This be- havior is due to the symmetry of the solar gradient around the Earth for variations of 180◦ of the Sun’s position.
If the orbit of the Earth had been considered perfectly
circular and the perturbation of the Sun approximated for small distances from the Earth, then every trajectory would remain unchanged if the position of the Sun was shifted by 180◦. Because there is a small eccentricity of the Earth orbit and the complete perturbation expression is considered, there are differences that require calculat- ing the trajectory even for the almost-symmetric cases;
however, it was possible to employ the solutions already computed as a starting guess for the corresponding 180◦ shifted cases.
As stated before, the inclinations of the obtained lunar orbits are free, and they result in a uniform distribution between 30◦and 165◦. The Horyu mission needs an equa- torial orbit around the Moon in order to properly achieve its objective; therefore, a constraint on the inclination of the lunar orbit is considered here. All the trajectories are forced to arrive in the equatorial orbit around the Moon; both prograde and retrograde equatorial orbits are considered valid, and the interval of inclinations allowed is i ∈ [0◦,30◦] and i∈ [150◦,180◦]. It was possible to achieve the orbits in this range for all trajectories—often at the cost of a highertf. Figure 10 shows the inclina- tions before the constraint (upper plot) and after the constraint (lower plot). Figure 11 shows the difference in time of flight, propellant, and ∆V with respect to
the unconstrained trajectories. It can be seen that for most of the trajectories the difference in duration is less than 10 days, and the maximum positive difference is 32.98 days—corresponding to 173.7 g of extra propellant and a ∆V of 93.22 m/s. In one case a notable difference is found: −24.85 days, −130.9 g, and−66.69 m/s.
To provide an outline of the trajectories not shown before, and to visualize how they vary with the variation of the angle parameter, twelve trajectories are plotted in Fig. 12 (one every 30◦)—around a visualization of the Sun–Earth–Moon angle in which the direction of the Moon is visualized as a blue arrow and the direction of the Sun for all the angles as red arrows. With this disposition, it is easy to see the zones in which typical trajectories appear. The trajectories in this figure include the constraint on the inclination.
5 Flyby sensitivity and effect of the solar radiation pressure
The initial flyby of the Moon has a significant influence on the rendezvous trajectory; further, the initial position of the Moon is of great importance in WSB transfers.
Here, a brief analysis of the sensitivity of the problem to small variations in the initial position of the Moon is
Fig. 10 Lunar inclination with and without equatorial constraint.
Fig. 11 Differences in time of flight, propellant mass, and total ∆V between trajectories with equatorial inclination of the final orbit and free trajectories.
Y Y
Y
Y
Y
Y
Y Y Y
Y Y Y
Z
ZZ
Z
Z
Z Z Z Z Z Z Z
152.5
122.7
63
33.2
5.8 92.9
Fig. 12 Outline of the transfer trajectories with theθsm0parameter. The blue arrow represents the direction of the Moon and the red arrows represent the direction of the Sun at twelve different angles; one angle is highlighted as an example.
Trajectories are plotted in the Earth-centered frame.
presented. The best trajectory among the ones found, in the case where the constraint on the inclination at the arrival at the Moon is considered as a case study;
the Moon is moved along its orbit by steps of 1◦ both ahead and behind the nominal initial position—then the trajectory is re-optimized to achieve the rendezvous again. With a displacement of −1◦, the performance worsens; the flyby has a lower altitude, and the duration of the trajectory is increased by 28.16 days. With a displacement of +1◦, the performance improved, the flyby height increased, and the duration was reduced by 4.64 days. Continuing the analysis in the favorable direction shows that the initial position displaced by +1◦ is a minimum around the nominal trajectory; in fact, moving to +2◦, the performance is still better—but of a smaller amount. At +3◦ instead, the duration increases, and also at +4◦ even when the shape of the trajectory is changed.
Table 4 summarizes what is said above, showing the dif- ference in duration, propellant, flyby height, and apogee with respect to the nominal trajectory. Figure 13 shows the trajectories found for every position variation and also the nominal trajectory (0◦) for comparison.
In the analysis in the previous section, the effect of the solar radiation pressure was not taken into account; here, we consider its effect on the best trajectory in a simplified way. The area of the spacecraft is considered constant during the trajectory, equal to the maximum area of the satellite and always facing the direction of the Sun.
Table 4 Results of the variation of the initial position of the Moon
Moon displ. δtf
(day)
δmp
(g)
δhflyby
(km)
δrapo
(105km)
−1◦ +28.16 +148.4 −979.7 +4.181
+1◦ −4.64 −24.5 +891.7 −0.706 +2◦ −4.08 −21.5 +1242.1 −0.430
+3◦ +2.40 +12.6 +1490.6 −0.481
+4◦ +37.84 +19.9 +3211.0 −4.175
Full specular reflection is considered in such a way that the acceleration given by the solar radiation pressure is always in the opposite direction of the Sun. These conditions allow an estimate of the maximum possible effect of the SRP on the trajectory.
aSRP=PEA
mrbs (10)
Equation (10) shows the expression of the acceleration of the SRP in the considered conditions;PE is the solar radiation pressure at 1 AU,Ais the area of the spacecraft, mis the mass, andrbs is the unit vector from the Sun to the spacecraft. The numerical results show that the solar radiation pressure has a negligible effect and the nominal trajectory has its duration changed by 2600 s.
6 Stabilization and circularization
In this section, the problem of obtaining a stable orbit around the Moon is addressed. The trajectory with the
Fig. 13 Transfer trajectory withθsm0=−5.78◦subject to small variations of the initial position of the Moon. Plots in the Earth-centred inertial reference frame.
least time of flight among the previous trajectories was considered as a case study.
The final semi-axis and eccentricity of the lunar ren- dezvous trajectory area= 29,571 km, e= 0.934. If left free, the spacecraft will quickly escape from the Moon;
instead, the spacecraft must achieve a circular orbit at an altitude of 100 km. For this purpose, the trajectory is continued with some arcs in which the direction of the thrust is constrained to vary in a cone around the anti-velocity direction; an optimization is run with the objective of reducing the semi-axis below 20,000 km and the eccentricity below 0.7, in order to reach a stable orbit.
Then, the orbit is circularized, and the height is reduced by employing the following strategy:
• First, the thrust is in the anti-velocity direction during the orbit when the true anomaly is inside an interval centered around the pericenter, here taken as [−130◦,130◦]. When this condition is not met, the thruster is shut off. This reduces the semi-axis and eccentricity of the orbit.
• After the eccentricity is reduced to almost zero, the thrust returns continuously during all the orbits in the anti-velocity direction until the required height is reached.
Table 5 shows the duration and propellant consumption of the considered trajectory; in the first two columns, the time and propellant are divided between the transfer trajectory (without the added arcs used for stabilization) and the stabilization and circularization maneuver until the final orbit is reached. The last column presents the total cost of the mission. The propellant needed is inside the 3 kg limit with a margin of 0.4928 kg, and the total duration is 765.77 days, of which 484.75 days are of thrusting. Figure 14 shows the trajectory from the arrival around the Moon to the end of the maneuver on the final orbit; and Fig. 15 shows the evolution of the orbital elements during stabilization. To be more readable, the plot starts from 81.54 days after departure from the Earth, when the original transfer trajectory is ended. The semi-axis is gradually reduced and the eccentricity is subject to some oscillations during the initial phase, before becoming more stable and starting to decrease until it reaches zero. The inclination of the orbit reached is 11◦—it is therefore acceptable for the imposed constraints.
As a final remark, if instead of a low orbit at an altitude of 100 km, the requirements had been to simply place the
Table 5 Cost of the trajectory from the Earth to low lunar orbit
Transfer trajectory (θsm0=−5.78◦)
Stabilization and
circularization Total
t(day) 81.54 684.23 765.77
mp(kg) 0.4297 2.0775 2.5072
Fig. 14 Trajectory to low lunar orbit, plotted in the Moon- centric inertial frame.
5
*&4
Fig. 15 Orbital elements during the stabilization and cir- cularization trajectory.
spacecraft in a stable orbit, then clearly the total cost would have been notably less. The trajectory presented in this section can be taken as an example; if it is truncated at 37.23 days after the end of the Earth–Moon transfer, it yields an orbit with a semi-axis of about 19,000 km (having a pericenter height of 6400 km, and an apocenter of 28,800 km) that remains stable for at least 500 days.
The propellant cost would be 0.2 kg over the transfer trajectory cost, so the total cost would be 0.626 kg—
about a quarter of the mass to reach the 100 km circular orbit.
7 Conclusions
This work presented the trajectory analysis of the Horyu- VI nano-satellite, an international lunar mission with the goal of studying the lunar horizon glow.
In particular, the effect of the solar gravity gradient on the trajectory was analyzed, covering the entire range of values of the Sun–Earth–Moon angle at departure. With the given low-thrust propulsion system, a transfer trajec- tory was found for all configurations, achieving a lunar orbit with eccentricity less than one, and a pericenter at an altitude of 200 km. The range of propellant required varies from 490 g to 5.47 kg, although the most expensive cases require a greater quantity of propellant than the expected 3 kg. It was found that the best trajectories resemble the WSB Earth–Moon transfers in the way they exploit the Sun gradient to decelerate and accelerate where most needed; therefore, these types of trajectories should be considered the most feasible for the considered problem.
A constraint on the inclination of the arrival orbit at the Moon is also considered, and the difference in cost and time is computed with respect to the unconstrained case. Reaching an equatorial orbit is possible in all configurations, and the differences in time and propellant are small.
Among the transfer trajectories, some further analysis was performed on the best one considering its sensitivity to small changes in the Moon’s initial position—which affect the flyby. It was found that a few degrees of variation in the angular position are admissible without drastic changes in the trajectory—and in some cases, they might even produce an improvement in the performance.
It was also assessed that the solar radiation pressure should be of very little importance for this mission.
The problem of the stabilization of the lunar orbit was also considered. In the specific case of the best transfer trajectory, the cost of this maneuver is higher than the transfer trajectory cost. Reaching the final low lunar orbit of the Horyu-VI mission required a total of 2.5 kg of propellant—equal to 12.5% of the initial mass. These results show that with the given allocated propellant
mass, and considering the best transfer trajectory, the mission results are feasible.
Funding note
Open Access funding provided by Universit`a degli Studi di Roma La Sapienza.
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Federico De Grossi received his M.S.
degree in space and astronautical engi- neering at Sapienza University of Rome in 2019, with a thesis on rendezvous tra- jectories with orbits around asteroids. He is currently a Ph.D. student, and his fields of study are space mission trajectory anal- ysis and optimization, optimization meth- ods, and quantum information and computation. E-mail:
federico.degrossi@uniroma1.it.
Paolo Marzioli is a post-doctoral re- search fellow at Sapienza University of Rome. He received his Ph.D. degree in aeronautical and space engineering in 2021, with a thesis on the implementation of navigation systems for nano-satellites.
His main research topics deal with nano- satellite missions, stratospheric experi- ment development and tracking, and navigation systems evaluation for different aerospace mission profiles. E-mail:
paolo.marzioli@uniroma1.it.
Mengu Cho received his B.S. and M.S.
degrees from the University of Tokyo, and his Ph.D. degree from Massachusetts Institute of Technology, in 1992. After working at Kobe University and Interna- tional Space University, he joined Kyushu Institute of Technology (Kyutech) in 1996. Since 2004, he has been a pro- fessor. Currently, he is the director of Laboratory of Lean Satellite Enterprises and In-Orbit Experiments. His research interests include spacecraft environmental interaction and satellite systems. He has supervised 11 university satellite projects, among which 9 projects and 16 satellites have al- ready been launched. He has authored or co-authored more than 180 papers in peer reviewed journals. His research interest is nano-/micro-satellite development and applica- tions. In 2019, he received Frank J. Malina Astronautics Medal from International Astronautical Federation. E-mail:
cho.mengu801@gmail.kyutech.jp.
Fabio Santoni is an associate pro- fessor of aerospace systems at Sapienza University of Rome. His main research topics are nano-satellite mission design, development and operations, space de- bris observations and mitigation, and sustainable space exploration. E-mail:
fabio.santoni@uniroma1.it.
Christian Circi is currently an asso- ciate professor in flight mechanics at the Department of Astronautical, Electrical, and Energy Engineering, Sapienza University of Rome. He received his M.S.
degree in aeronautical engineering and aerospace engineering and his Ph.D. de- gree in aerospace engineering at Sapienza University of Rome. He worked as a researcher at the Grupo de Mecanica of Vuelo-Madrid (GMV) and a research assistant at the Department of Aerospace Engineering. He is a lecturer in “Interplanetary Trajectories” and “Flight Mechanics of Launcher” in the master degree course of space and astronautical engineering at Sapienza University of Rome. His principal research fields are third-body and solar perturbations, interplanetary and lunar trajectories, so-
lar sails, orbits for planetary observation, and the as- cent trajectory of Launcher. He is the associate editor forAerospace Science and Technology and the Interna- tional Journal of Aerospace Engineering. E-mail: chris- tian.circi@uniroma1.it.
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